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Theorem ntrclscls00 41676
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)
Hypotheses
Ref Expression
ntrcls.o 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
ntrcls.d 𝐷 = (𝑂𝐵)
ntrcls.r (𝜑𝐼𝐷𝐾)
Assertion
Ref Expression
ntrclscls00 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐼,𝑘   𝑗,𝐾,𝑘   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝐼(𝑖)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclscls00
StepHypRef Expression
1 ntrcls.o . . . . . 6 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2 ntrcls.d . . . . . 6 𝐷 = (𝑂𝐵)
3 ntrcls.r . . . . . 6 (𝜑𝐼𝐷𝐾)
41, 2, 3ntrclsfv1 41665 . . . . 5 (𝜑 → (𝐷𝐼) = 𝐾)
54fveq1d 6776 . . . 4 (𝜑 → ((𝐷𝐼)‘∅) = (𝐾‘∅))
62, 3ntrclsbex 41644 . . . . 5 (𝜑𝐵 ∈ V)
71, 2, 3ntrclsiex 41663 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
8 eqid 2738 . . . . 5 (𝐷𝐼) = (𝐷𝐼)
9 0elpw 5278 . . . . . 6 ∅ ∈ 𝒫 𝐵
109a1i 11 . . . . 5 (𝜑 → ∅ ∈ 𝒫 𝐵)
11 eqid 2738 . . . . 5 ((𝐷𝐼)‘∅) = ((𝐷𝐼)‘∅)
121, 2, 6, 7, 8, 10, 11dssmapfv3d 41627 . . . 4 (𝜑 → ((𝐷𝐼)‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
135, 12eqtr3d 2780 . . 3 (𝜑 → (𝐾‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
14 dif0 4306 . . . . . . 7 (𝐵 ∖ ∅) = 𝐵
1514fveq2i 6777 . . . . . 6 (𝐼‘(𝐵 ∖ ∅)) = (𝐼𝐵)
16 id 22 . . . . . 6 ((𝐼𝐵) = 𝐵 → (𝐼𝐵) = 𝐵)
1715, 16eqtrid 2790 . . . . 5 ((𝐼𝐵) = 𝐵 → (𝐼‘(𝐵 ∖ ∅)) = 𝐵)
1817difeq2d 4057 . . . 4 ((𝐼𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = (𝐵𝐵))
19 difid 4304 . . . 4 (𝐵𝐵) = ∅
2018, 19eqtrdi 2794 . . 3 ((𝐼𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = ∅)
2113, 20sylan9eq 2798 . 2 ((𝜑 ∧ (𝐼𝐵) = 𝐵) → (𝐾‘∅) = ∅)
22 pwidg 4555 . . . . 5 (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
236, 22syl 17 . . . 4 (𝜑𝐵 ∈ 𝒫 𝐵)
241, 2, 3, 23ntrclsfv 41669 . . 3 (𝜑 → (𝐼𝐵) = (𝐵 ∖ (𝐾‘(𝐵𝐵))))
2519fveq2i 6777 . . . . . 6 (𝐾‘(𝐵𝐵)) = (𝐾‘∅)
26 id 22 . . . . . 6 ((𝐾‘∅) = ∅ → (𝐾‘∅) = ∅)
2725, 26eqtrid 2790 . . . . 5 ((𝐾‘∅) = ∅ → (𝐾‘(𝐵𝐵)) = ∅)
2827difeq2d 4057 . . . 4 ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵𝐵))) = (𝐵 ∖ ∅))
2928, 14eqtrdi 2794 . . 3 ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵𝐵))) = 𝐵)
3024, 29sylan9eq 2798 . 2 ((𝜑 ∧ (𝐾‘∅) = ∅) → (𝐼𝐵) = 𝐵)
3121, 30impbida 798 1 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2106  Vcvv 3432  cdif 3884  c0 4256  𝒫 cpw 4533   class class class wbr 5074  cmpt 5157  cfv 6433  (class class class)co 7275  m cmap 8615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617
This theorem is referenced by: (None)
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